Properties

Label 2-51600-1.1-c1-0-26
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 5·11-s + 3·13-s − 17-s + 4·19-s + 4·21-s − 7·23-s − 27-s + 7·31-s − 5·33-s + 4·37-s − 3·39-s − 9·41-s + 43-s + 9·49-s + 51-s − 3·53-s − 4·57-s + 8·59-s − 4·61-s − 4·63-s + 67-s + 7·69-s + 12·71-s − 20·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s − 1.45·23-s − 0.192·27-s + 1.25·31-s − 0.870·33-s + 0.657·37-s − 0.480·39-s − 1.40·41-s + 0.152·43-s + 9/7·49-s + 0.140·51-s − 0.412·53-s − 0.529·57-s + 1.04·59-s − 0.512·61-s − 0.503·63-s + 0.122·67-s + 0.842·69-s + 1.42·71-s − 2.27·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.657268759\)
\(L(\frac12)\) \(\approx\) \(1.657268759\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
43 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.36448046729030, −13.83760677743243, −13.51652452078853, −12.96891547715750, −12.23277133076480, −12.03059593442319, −11.51689513483478, −10.95664387024862, −10.23262017579600, −9.729796020782137, −9.540427191510648, −8.812872487634597, −8.266159714918286, −7.552430149418869, −6.742065859425639, −6.491068897801101, −6.130103491089743, −5.516915897760608, −4.693922254950877, −3.932040636425093, −3.647012212360758, −2.978513175380696, −2.036369502871696, −1.188443852775662, −0.5238468654081729, 0.5238468654081729, 1.188443852775662, 2.036369502871696, 2.978513175380696, 3.647012212360758, 3.932040636425093, 4.693922254950877, 5.516915897760608, 6.130103491089743, 6.491068897801101, 6.742065859425639, 7.552430149418869, 8.266159714918286, 8.812872487634597, 9.540427191510648, 9.729796020782137, 10.23262017579600, 10.95664387024862, 11.51689513483478, 12.03059593442319, 12.23277133076480, 12.96891547715750, 13.51652452078853, 13.83760677743243, 14.36448046729030

Graph of the $Z$-function along the critical line